Classifying rendezvous tasks of arbitrary dimension

The rendezvous is a type of distributed decision tasks including many well-known tasks such as set agreement, simplex agreement, and approximation agreement. An n-dimensional rendezvous task, n≥1, allows n+2 distinct input values, and each execution produces at most n+2 distinct output values. A rendezvous task is said to implement another if an instance of its solution, followed by a protocol based on shared read/write registers, solves the other. The notion of implementation induces a classification of rendezvous tasks of every dimension: two tasks belong to the same class if they implement each other. Previous work on classifying rendezvous tasks only focused on 1-dimensional ones.This paper solves an open problem by presenting the classification of nice rendezvous of arbitrary dimension. An n-dimensional rendezvous task is said to be nice if the qth reduced homology group of its decision space is trivial for q≠n, and free for q=n. Well-known examples are set agreement, simplex agreement, and approximation agreement. Each n-dimensional rendezvous task is assigned an algebraic signature, which consists of the nth homology group of the decision space, as well as a distinguished element in the group. It is shown that an n-dimensional nice rendezvous task implements another if and only if there is a homomorphism from its signature to that of the other. Hence the computational power of a nice rendezvous task is completely characterized by its signature.In each dimension, there are infinitely many classes of rendezvous tasks, and exactly countable classes of nice ones. A representative is explicitly constructed for each class of nice rendezvous tasks.